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Theorem pythagtriplem14 16799

Description: Lemma for pythagtrip 16805. Calculate the square of 𝑁. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

**Hypothesis**
Ref	Expression
pythagtriplem13.1	⊢ 𝑁 = (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2)

**Assertion**
Ref	Expression
pythagtriplem14	⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝑁↑2) = ((𝐶 − 𝐴) / 2))

**Proof of Theorem pythagtriplem14**
Step	Hyp	Ref	Expression
1		pythagtriplem13.1	. . 3 ⊢ 𝑁 = (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2)
2	1	oveq1i 7377	. 2 ⊢ (𝑁↑2) = ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2)↑2)
3		nncn 12182	. . . . . . . . 9 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℂ)
4		nncn 12182	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
5		addcl 11120	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 + 𝐵) ∈ ℂ)
6	3, 4, 5	syl2anr 598	. . . . . . . 8 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℂ)
7	6	sqrtcld 15402	. . . . . . 7 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (√‘(𝐶 + 𝐵)) ∈ ℂ)
8		subcl 11392	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 − 𝐵) ∈ ℂ)
9	3, 4, 8	syl2anr 598	. . . . . . . 8 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℂ)
10	9	sqrtcld 15402	. . . . . . 7 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (√‘(𝐶 − 𝐵)) ∈ ℂ)
11	7, 10	subcld 11505	. . . . . 6 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) ∈ ℂ)
12	11	3adant1 1131	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) ∈ ℂ)
13	12	3ad2ant1 1134	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) ∈ ℂ)
14		2cn 12256	. . . . 5 ⊢ 2 ∈ ℂ
15		2ne0 12285	. . . . 5 ⊢ 2 ≠ 0
16		sqdiv 14083	. . . . 5 ⊢ ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2)↑2) = ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / (2↑2)))
17	14, 15, 16	mp3an23 1456	. . . 4 ⊢ (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) ∈ ℂ → ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2)↑2) = ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / (2↑2)))
18	13, 17	syl 17	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2)↑2) = ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / (2↑2)))
19	14	sqvali 14142	. . . . 5 ⊢ (2↑2) = (2 · 2)
20	19	oveq2i 7378	. . . 4 ⊢ ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / (2↑2)) = ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / (2 · 2))
21	13	sqcld 14106	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) ∈ ℂ)
22		2cnne0 12386	. . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
23		divdiv1 11866	. . . . . . 7 ⊢ (((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / 2) / 2) = ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / (2 · 2)))
24	22, 22, 23	mp3an23 1456	. . . . . 6 ⊢ ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) ∈ ℂ → (((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / 2) / 2) = ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / (2 · 2)))
25	21, 24	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / 2) / 2) = ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / (2 · 2)))
26		simp12 1206	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℕ)
27		simp13 1207	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℕ)
28	26, 27, 7	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 + 𝐵)) ∈ ℂ)
29	26, 27, 10	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 − 𝐵)) ∈ ℂ)
30		binom2sub 14182	. . . . . . . . . 10 ⊢ (((√‘(𝐶 + 𝐵)) ∈ ℂ ∧ (√‘(𝐶 − 𝐵)) ∈ ℂ) → (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) = ((((√‘(𝐶 + 𝐵))↑2) − (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) + ((√‘(𝐶 − 𝐵))↑2)))
31	28, 29, 30	syl2anc 585	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) = ((((√‘(𝐶 + 𝐵))↑2) − (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) + ((√‘(𝐶 − 𝐵))↑2)))
32		nnre 12181	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℝ)
33		nnre 12181	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
34		readdcl 11121	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 + 𝐵) ∈ ℝ)
35	32, 33, 34	syl2anr 598	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℝ)
36	35	3adant1 1131	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℝ)
37	36	3ad2ant1 1134	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℝ)
38	37	recnd 11173	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℂ)
39		resubcl 11458	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 − 𝐵) ∈ ℝ)
40	32, 33, 39	syl2anr 598	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℝ)
41	40	3adant1 1131	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℝ)
42	41	3ad2ant1 1134	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 − 𝐵) ∈ ℝ)
43	42	recnd 11173	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 − 𝐵) ∈ ℂ)
44	7	3adant1 1131	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (√‘(𝐶 + 𝐵)) ∈ ℂ)
45	10	3adant1 1131	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (√‘(𝐶 − 𝐵)) ∈ ℂ)
46	44, 45	mulcld 11165	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))) ∈ ℂ)
47		mulcl 11122	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))) ∈ ℂ) → (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵)))) ∈ ℂ)
48	14, 46, 47	sylancr 588	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵)))) ∈ ℂ)
49	48	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵)))) ∈ ℂ)
50	38, 43, 49	addsubd 11526	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶 + 𝐵) + (𝐶 − 𝐵)) − (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) = (((𝐶 + 𝐵) − (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) + (𝐶 − 𝐵)))
51	27	nncnd 12190	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℂ)
52		simp11 1205	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℕ)
53	52	nncnd 12190	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℂ)
54		subdi 11583	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (2 · (𝐶 − 𝐴)) = ((2 · 𝐶) − (2 · 𝐴)))
55	14, 51, 53, 54	mp3an2i 1469	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · (𝐶 − 𝐴)) = ((2 · 𝐶) − (2 · 𝐴)))
56		ppncan 11436	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (𝐶 + 𝐶))
57	56	3anidm13 1423	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (𝐶 + 𝐶))
58		2times 12312	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 ∈ ℂ → (2 · 𝐶) = (𝐶 + 𝐶))
59	58	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2 · 𝐶) = (𝐶 + 𝐶))
60	57, 59	eqtr4d 2774	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (2 · 𝐶))
61	3, 4, 60	syl2anr 598	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (2 · 𝐶))
62	61	3adant1 1131	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (2 · 𝐶))
63	62	3ad2ant1 1134	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (2 · 𝐶))
64	26	nncnd 12190	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℂ)
65		subsq 14172	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶↑2) − (𝐵↑2)) = ((𝐶 + 𝐵) · (𝐶 − 𝐵)))
66	51, 64, 65	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶↑2) − (𝐵↑2)) = ((𝐶 + 𝐵) · (𝐶 − 𝐵)))
67		oveq1 7374	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) → (((𝐴↑2) + (𝐵↑2)) − (𝐵↑2)) = ((𝐶↑2) − (𝐵↑2)))
68	67	3ad2ant2 1135	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐴↑2) + (𝐵↑2)) − (𝐵↑2)) = ((𝐶↑2) − (𝐵↑2)))
69		nncn 12182	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
70	69	sqcld 14106	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ ℕ → (𝐴↑2) ∈ ℂ)
71	70	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴↑2) ∈ ℂ)
72	4	sqcld 14106	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐵 ∈ ℕ → (𝐵↑2) ∈ ℂ)
73	72	3ad2ant2 1135	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐵↑2) ∈ ℂ)
74	71, 73	pncand 11506	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (((𝐴↑2) + (𝐵↑2)) − (𝐵↑2)) = (𝐴↑2))
75	74	3ad2ant1 1134	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐴↑2) + (𝐵↑2)) − (𝐵↑2)) = (𝐴↑2))
76	68, 75	eqtr3d 2773	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶↑2) − (𝐵↑2)) = (𝐴↑2))
77	66, 76	eqtr3d 2773	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 + 𝐵) · (𝐶 − 𝐵)) = (𝐴↑2))
78	77	fveq2d 6844	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘((𝐶 + 𝐵) · (𝐶 − 𝐵))) = (√‘(𝐴↑2)))
79	32	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈ ℝ)
80	33	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈ ℝ)
81		nngt0 12208	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐶 ∈ ℕ → 0 < 𝐶)
82	81	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < 𝐶)
83		nngt0 12208	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵)
84	83	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < 𝐵)
85	79, 80, 82, 84	addgt0d 11725	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 < (𝐶 + 𝐵))
86		0re 11146	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ∈ ℝ
87		ltle 11234	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ ℝ ∧ (𝐶 + 𝐵) ∈ ℝ) → (0 < (𝐶 + 𝐵) → 0 ≤ (𝐶 + 𝐵)))
88	86, 87	mpan 691	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 + 𝐵) ∈ ℝ → (0 < (𝐶 + 𝐵) → 0 ≤ (𝐶 + 𝐵)))
89	35, 85, 88	sylc 65	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 ≤ (𝐶 + 𝐵))
90	89	3adant1 1131	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 ≤ (𝐶 + 𝐵))
91	90	3ad2ant1 1134	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 ≤ (𝐶 + 𝐵))
92		pythagtriplem10 16791	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → 0 < (𝐶 − 𝐵))
93	92	3adant3 1133	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 < (𝐶 − 𝐵))
94		ltle 11234	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ (𝐶 − 𝐵) ∈ ℝ) → (0 < (𝐶 − 𝐵) → 0 ≤ (𝐶 − 𝐵)))
95	86, 94	mpan 691	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 − 𝐵) ∈ ℝ → (0 < (𝐶 − 𝐵) → 0 ≤ (𝐶 − 𝐵)))
96	42, 93, 95	sylc 65	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 ≤ (𝐶 − 𝐵))
97	37, 91, 42, 96	sqrtmuld 15387	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘((𝐶 + 𝐵) · (𝐶 − 𝐵))) = ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))
98	78, 97	eqtr3d 2773	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐴↑2)) = ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))
99		nnre 12181	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
100	99	3ad2ant1 1134	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℝ)
101	100	3ad2ant1 1134	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℝ)
102		nnnn0 12444	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ₀)
103	102	nn0ge0d 12501	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℕ → 0 ≤ 𝐴)
104	103	3ad2ant1 1134	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 ≤ 𝐴)
105	104	3ad2ant1 1134	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 ≤ 𝐴)
106	101, 105	sqrtsqd 15382	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐴↑2)) = 𝐴)
107	98, 106	eqtr3d 2773	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))) = 𝐴)
108	107	oveq2d 7383	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵)))) = (2 · 𝐴))
109	63, 108	oveq12d 7385	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶 + 𝐵) + (𝐶 − 𝐵)) − (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) = ((2 · 𝐶) − (2 · 𝐴)))
110	55, 109	eqtr4d 2774	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · (𝐶 − 𝐴)) = (((𝐶 + 𝐵) + (𝐶 − 𝐵)) − (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))))
111		resqrtth 15217	. . . . . . . . . . . . 13 ⊢ (((𝐶 + 𝐵) ∈ ℝ ∧ 0 ≤ (𝐶 + 𝐵)) → ((√‘(𝐶 + 𝐵))↑2) = (𝐶 + 𝐵))
112	37, 91, 111	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵))↑2) = (𝐶 + 𝐵))
113	112	oveq1d 7382	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((√‘(𝐶 + 𝐵))↑2) − (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) = ((𝐶 + 𝐵) − (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))))
114		resqrtth 15217	. . . . . . . . . . . 12 ⊢ (((𝐶 − 𝐵) ∈ ℝ ∧ 0 ≤ (𝐶 − 𝐵)) → ((√‘(𝐶 − 𝐵))↑2) = (𝐶 − 𝐵))
115	42, 96, 114	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 − 𝐵))↑2) = (𝐶 − 𝐵))
116	113, 115	oveq12d 7385	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵))↑2) − (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) + ((√‘(𝐶 − 𝐵))↑2)) = (((𝐶 + 𝐵) − (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) + (𝐶 − 𝐵)))
117	50, 110, 116	3eqtr4rd 2782	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵))↑2) − (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) + ((√‘(𝐶 − 𝐵))↑2)) = (2 · (𝐶 − 𝐴)))
118	31, 117	eqtrd 2771	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) = (2 · (𝐶 − 𝐴)))
119	118	oveq1d 7382	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / 2) = ((2 · (𝐶 − 𝐴)) / 2))
120		subcl 11392	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐶 − 𝐴) ∈ ℂ)
121	3, 69, 120	syl2anr 598	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐴) ∈ ℂ)
122	121	3adant2 1132	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐴) ∈ ℂ)
123	122	3ad2ant1 1134	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 − 𝐴) ∈ ℂ)
124		divcan3 11835	. . . . . . . . 9 ⊢ (((𝐶 − 𝐴) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → ((2 · (𝐶 − 𝐴)) / 2) = (𝐶 − 𝐴))
125	14, 15, 124	mp3an23 1456	. . . . . . . 8 ⊢ ((𝐶 − 𝐴) ∈ ℂ → ((2 · (𝐶 − 𝐴)) / 2) = (𝐶 − 𝐴))
126	123, 125	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((2 · (𝐶 − 𝐴)) / 2) = (𝐶 − 𝐴))
127	119, 126	eqtrd 2771	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / 2) = (𝐶 − 𝐴))
128	127	oveq1d 7382	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / 2) / 2) = ((𝐶 − 𝐴) / 2))
129	25, 128	eqtr3d 2773	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / (2 · 2)) = ((𝐶 − 𝐴) / 2))
130	20, 129	eqtrid 2783	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))↑2) / (2↑2)) = ((𝐶 − 𝐴) / 2))
131	18, 130	eqtrd 2771	. 2 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2)↑2) = ((𝐶 − 𝐴) / 2))
132	2, 131	eqtrid 2783	1 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝑁↑2) = ((𝐶 − 𝐴) / 2))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 class class class wbr 5085 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 < clt 11179 ≤ cle 11180 − cmin 11377 / cdiv 11807 ℕcn 12174 2c2 12236 ↑cexp 14023 √csqrt 15195 ∥ cdvds 16221 gcd cgcd 16463

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-sup 9355 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-z 12525 df-uz 12789 df-rp 12943 df-seq 13964 df-exp 14024 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198

This theorem is referenced by: pythagtriplem15 16800 pythagtriplem17 16802

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